Structural Health Monitoring http://shm.sagepub.com/ WaveFormRevealer: An analytical framework and predictive tool for the simulation of multi-modal guided wave propagation and interaction with damage Yanfeng Shen and Victor Giurgiutiu Structural Health Monitoring published online 13 May 2014 DOI: 10.1177/1475921714532986 The online version of this article can be found at: http://shm.sagepub.com/content/early/2014/05/12/1475921714532986 Published by: http://www.sagepublications.com Additional services and information for Structural Health Monitoring can be found at: Email Alerts: http://shm.sagepub.com/cgi/alerts Subscriptions: http://shm.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://shm.sagepub.com/content/early/2014/05/12/1475921714532986.refs.html >> OnlineFirst Version of Record - May 13, 2014 What is This? Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Original Article WaveFormRevealer: An analytical framework and predictive tool for the simulation of multi-modal guided wave propagation and interaction with damage Structural Health Monitoring 1–21 Ó The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1475921714532986 shm.sagepub.com Yanfeng Shen and Victor Giurgiutiu Abstract This article presents the WaveFormRevealer—an analytical framework and predictive tool for the simulation of guided Lamb wave propagation and interaction with damage. The theory of inserting damage effects into the analytical model is addressed, including wave transmission, reflection, mode conversion, and nonlinear higher harmonics. The analytical model is coded into MATLAB, and a graphical user interface (WaveFormRevealer graphical user interface) is developed to obtain real-time predictive waveforms for various combinations of sensors, structural properties, and damage. In this article, the main functions of WaveFormRevealer are introduced. Case studies of selective Lamb mode linear and nonlinear interaction with damage are presented. Experimental verifications are carried out. The article finishes with summary and conclusions followed by recommendations for further work. Keywords Guided waves, structural health monitoring, damage detection, piezoelectric wafer active sensors, analytical model, nonlinear ultrasonics Introduction Guided waves retain a central function in the development of structural health monitoring (SHM) systems using piezoelectric wafer active sensor (PWAS) principles. The modeling of Lamb waves is challenging, because Lamb waves propagate in structures with multi-mode dispersive characteristics. At a certain value of the plate thickness-frequency product, several Lamb modes may exist simultaneously, and their phase velocities vary with frequency.1–3 When Lamb waves interact with damage, they will be transmitted, reflected, scattered, and mode converted. Nonlinear interaction with damage may also exist, and this will introduce distinctive features like higher harmonics.4–6 These aspects give rise to the complexity of modeling the interaction between Lamb waves and damage. To solve such complicated problems, numerical methods like finite element method (FEM) and boundary element method (BEM) are usually adopted. However, to ensure the accuracy of simulating high-frequency waves of short wavelengths, the transient analysis requires considerably small time step and very fine mesh (T =Dt; l=lFEM 20 ; 30), which is expensive both in computational time and computer resources.7,8 Analytical model provides an alternative approach to attack the same problem with much less cost.9 PWAS transducers are a convenient way of transmitting and receiving guided waves in structures for SHM applications.3 The analytical model of PWAS-generated Lamb waves and its tuning effect has been investigated, and a close-form solution for straight crested guided Lamb wave was derived by Giurgiutiu.10 Extension of tuning concepts to 2D analytical models of Lamb waves generated by finite-dimensional piezoelectric transducers was given in Raghavan and Cesnik.11 These analytical developments facilitate the understanding of PWAS-coupled Lamb waves for SHM applications. Department of Mechanical Engineering, University of South Carolina, Columbia, SC, USA Corresponding author: Yanfeng Shen, Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA. Email: [email protected] Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 2 Structural Health Monitoring However, these analytical solutions only applied to guided wave propagation in pristine structures, whereas the use of Lamb waves in SHM applications requires that their interaction with damage also be studied. After interacting with damage, Lamb waves will carry damage information resulting in waveforms with special characteristics (phase change, new wave packets generation through mode conversion, higher harmonic components, etc.), which need to be investigated for damage detection. Several investigators have studied the interaction between guided waves and damage analytically using normal-mode expansion and boundary-condition matching.12–18 Damage interaction coefficients were derived to quantify the guided wave transmission, reflection, mode conversion, and scatter at the damage site. Due to their mathematical complexity, these analytical solutions are restricted to simple damage geometries: notches, holes, and partially through holes. Extension to more complicated damage geometries has been attempted through series expansion of the rugged damage contour. In the generic case of arbitrary-shape damage, the numerical approaches using space discretization (FEM, BEM) are used due to their convenience, but on the expense of orders of magnitude increase in computational time and/or computer resources. The design of a SHM system requires computationally efficient predictive tools that permit the exploration of a wide parameter space to identify the optimal combination between the transducers type, size, number, and guided wave characteristics (mode type, frequency, and wavelength) to achieve best detection and quantification of a certain damage type. Such parameter space exploration desiderate can be best achieved with analytical tools which are fast and efficient. In this article, we describe an analytical approach based on the one-dimensional (1D) (straight crested) guided wave propagation analysis. In our study, we inserted the damage effect into the analytical model by considering wave transmission, reflection, mode conversion, and higher harmonics components described through damage interaction coefficients at the damage site. We do not attempt to derive these damage interaction coefficients here, but assume that they are available either from literature or from FEM, BEM analysis performed separately in a separate computational module. This analytical approach was coded into MATLAB and the WaveFormRevealer (WFR) graphical user interface (GUI) was developed. The WFR can generate fast predictions of waveforms resulting from Lamb wave interaction with damage for arbitrary positioning of PWAS transmitters and receivers with respect to damage and with respect to each other. The users may choose their own excitation signal, PWAS size, structural parameters, and damage description. The current version of the WFR code is limited to 1D (straight crested) guided wave propagation; extension of this approach to two-dimensional (2D) (circular crested) guided wave propagation will be attempted in the future. PWAS Fundamentals PWAS couple the electrical and mechanical effects (mechanical strain, Sij ; mechanical stress, Tkl ; electrical field, Ek ; and electrical displacement, Dj ) through the tensorial piezoelectric constitutive equations Sij ¼ sEijkl Tkl þ dkij Ek Dj ¼ djkl Tkl þ eTjk Ek ð1Þ where sEijkl is the mechanical compliance of the material measured at zero electric field (E ¼ 0), eTjk is the dielectric permittivity measured at zero mechanical stress (T ¼ 0), and djkl represents the piezoelectric coupling effect. PWAS utilize the d31 coupling between in-plane strains, S1 ; S2 , and transverse electric field E3 . PWAS transducers can be used as both transmitters and receivers. Their modes of operation are shown in Figure 1. PWAS can serve several purposes3: (a) highbandwidth strain sensors, (b) high-bandwidth wave exciters and receivers, (c) resonators, and (d) embedded modal sensors with the electromechanical (E/M) impedance method. By application types, PWAS transducers can be used for (a) active sensing of far-field damage using pulse-echo, pitch-catch, and phased-array methods, (b) active sensing of near-field damage using highfrequency E/M impedance method and thickness gage mode, and (c) passive sensing of damage-generating events through detection of low-velocity impacts and acoustic emission at the tip of advancing cracks (Figure 1). The main advantage of PWAS over conventional ultrasonic probes is in their small size, lightweight, low profile, and small cost. In spite of their small size, PWAS are able to replicate many of the functions performed by conventional ultrasonic probes. Analytical modeling of Lamb waves interacting with damage Analytical modeling of guided Lamb waves propagation in a pristine structure One aspect of the difficulties in modeling Lamb wave propagation is due to their multi-mode feature. WFR is capable of modeling multi-mode Lamb wave propagation in structures. From Rayleigh–Lamb equation, it is found that the existence of certain Lamb mode depends on the plate thickness-frequency product. The fundamental S0 and A0 modes will always exist, but Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Shen and Giurgiutiu 3 Figure 1. Schematic of PWAS application modes (from Giurgiutiu19): (a) propagating Lamb waves, (b) standing Lamb waves (E/M impedance), and (c) PWAS phased arrays. E/M: electromechanical; PWAS: piezoelectric wafer active sensor; AE: acoustic emission. Transmitter PWAS Excitation: VT (ω ) Receiver PWAS Signal: VR ( xr , ω ) x=0 x=xr T-PWAS Piezoelectric transduction Elec.→Mech. R-PWAS Ultrasonic guided waves from T-PWAS undergo dispersion according to structural transfer function G ( xr , ω ) Piezoelectric transduction Mech.→Elec. Figure 2. A pitch-catch configuration between a transmitter PWAS (T-PWAS) and a receiver PWAS. PWAS: piezoelectric wafer active sensor. the higher modes will only appear beyond the cutoff frequencies.1 This section describes how an electrical tone burst applied to a transmitter PWAS (T-PWAS) transducer propagates through a structural waveguide to the receiver PWAS (R-PWAS) transducer in pitch-catch mode (Figure 2). The propagation takes place through ultrasonic guided Lamb waves which are generated at the T-PWAS through piezoelectric transduction and then captured and converted back into electric signal at the R-PWAS. Since several Lamb wave modes traveling with different wave speeds exist simultaneously, the electrical tone-burst applied on the T-PWAS will generate several wave packets. These wave packets will travel independently through the waveguide and will arrive at different times at the R-PWAS where they are converted back into electric signals through piezoelectric transduction. The predictive analytical model for Lamb wave propagation between the T-PWAS and RPWAS is constructed in frequency domain in the following steps (Figure 3(a)). Step 1. Perform Fourier transform of the time-domain excitation signal VT ðtÞ to obtain the frequency-domain ~T ðvÞ. For a tone burst, the signal excitation spectrum, V ~T ðvÞ are shown in VT ðtÞ and its Fourier transform V Figure 4. Step 2. Calculate the frequency-domain structural transfer function Gðxr ; vÞ from T-PWAS to R-PWAS. The structure transfer function Gðxr ; vÞ is given in equation (99) in the study by Giurgiutiu,3 page 327, which gives the in-plane wave strain at the plate surface as 8 NS jS iðjS xvtÞ at 0 <X S ex ðx; tÞ ¼ i ðsin j aÞ 0 S e m : S DS j j ) A X NA j iðjA xvtÞ A þ ðsin j aÞ 0 A e DA j jA Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 ð2Þ 4 Structural Health Monitoring (a) (b) Flow chart for prisne case guided wave propagaon Flow chart for guided wave propagaon with damage interacon Fourier transform of excitaon signal Fourier transform of excitaon signal FFT VT ( t ) → VT (ω ) VT ( t ) → VT (ω ) Calculaon of structural transfer funcon Calculaon of structural transfer funcon up to damage locaon G ( xr , ω ) G ( xd , ω ) Mulply excitaon with structural transfer funcon Mulply excitaon with structural transfer funcon to the damage locaon FFT VD ( xd , ω ) = G ( xd , ω )iVT (ω ) VR ( xr , ω ) = G ( xr , ω )iVT (ω ) Transmission, reﬂecon, mode conversion, and higher harmonics New wave source VNewSource ( xd , ω ) This new wave source is mulplied with structural transfer funcon from damage upto R-PWAS VR ( xd , xr , ω ) = G ( xr − xd , ω )iVNewSource ( xd , ω ) Perform inverse Fourier transform { } VT ( xr , t ) = IFFT VR ( xr , ω ) Perform inverse Fourier transform VT ( xd , xr , t ) = IFFT {VR ( xd , xr , ω )} Figure 3. WaveFormRevealer flow charts: (a) propagation in a pristine structural waveguide and (b) propagation and interaction with damage at location xd . where j is the frequency-dependent wavenumber of each Lamb wave mode and the superscripts S and A refer to symmetric and antisymmetric Lamb wave modes. If only the two fundamental modes, S0 and A0, are present, then Gðxr ; vÞ can be written as Gðxr ; vÞ ¼ S ðvÞeij S þ AðvÞeij S S NS j S ðvÞ ¼ kPWAS sin j a 0 S ; D S j NA jA AðvÞ ¼ kPWAS sin jA a 0 A DA j Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 xr A xr ð3Þ Shen and Giurgiutiu 5 Figure 4. Tone burst signal: (a) time domain and (b) frequency domain (From Giurgiutiu,3 p. 153). where Ns ðjÞ ¼ jb j2 þ b2 cos ad cos bd; 2 Ds ¼ j2 b2 cos ad sin bd þ 4j2 ab sin ad cos bd NA ðjÞ ¼ jb j2 þ b2 sin ad sin bd; 2 DA ¼ j2 b2 sin ad cos bd þ 4j2 ab cos ad sin bd sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v2 v2 l þ 2m 2 2 2 2 ; a ¼ 2 j ; b ¼ 2 j ; cp ¼ r cp cs rﬃﬃﬃﬃ m at 0 ð4Þ cs ¼ ; kPWAS ¼ i r m kPWAS is the complex transduction coefficient that converts applied voltage into guided wave strain at the T-PWAS, a is half length of PWAS size, and d is plate half thickness. The modal participation functions SðvÞ and AðvÞ determine the amplitudes of the S0 and A0 wave modes. The terms sinðjS aÞ and sinðjA aÞ control the tuning between the PWAS transducer and the Lamb waves. l and m are Lame’s constants of the structural material, and r is the material density. The wavenumber j of a specific mode for certain frequency v is calculated from Rayleigh–Lamb equation " #61 tan bd 4abj2 ¼ 2 tan ad j 2 b2 ð5Þ where +1 exponent corresponds to symmetric Lamb wave modes and 21 exponent corresponds to antisymmetric Lamb wave modes. Step 3. Multiply the structural transfer function by frequency-domain excitation signal (Figure 4(b)) to obtain the frequency-domain signal at the R-PWAS, Figure 5. (a) T-PWAS signal and (b) R-PWAS signal. T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor. ~R ðxr ; vÞ ¼ Gðxr ; vÞ V ~T ðvÞ. Hence, the wave that is, V arriving at the R-PWAS location is ~R ðxr ; vÞ ¼ S ðvÞV ~T ðvÞeijS xr þ AðvÞV ~T ðvÞeijA xr V ð6Þ This signal in equation (6) can be decomposed into symmetric and antisymmetric components ~ S ðxr ; vÞ ¼ S ðvÞV ~T ðvÞeijS xr V R ð7Þ ~ A ðxr ; vÞ ¼ AðvÞV ~T ðvÞeijA xr V R ð8Þ Step 4. Perform the inverse Fourier transform to obtain the time-domain wave signal at the R-PWAS, that is VR ðxr ; tÞ ¼ IFFT fV~R ðxr ; vÞg ð9Þ Due to the multi-mode character of guided Lamb wave propagation, the received signal has at least two separate wave packets, S0 and A0 (Figure 5). This analysis can be extended to include higher guided wave modes (S1, A1, etc.), that is Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 6 Structural Health Monitoring Receiver PWAS Signal: VR ( xd , xr , ω ) Transmitter PWAS Excitation: VT (ω ) x=0 x = xr x = xd Damage T-PWAS R-PWAS Ultrasonic guided waves from TPiezoelectric PWAS transduction Elec.→Mech Transfer function . G ( xd , ω ) Nonlinear damage: new wave source Transmission, reflection, mode conversion, and higher harmonics Ultrasonic waves with damage information Transfer function G ( xr − xd , ω ) Piezoelectric transduction Mech.→Elec. Figure 6. A pitch-catch configuration between a transmitter PWAS and a receiver PWAS. PWAS: receiver piezoelectric wafer active sensor. ~R ðxr ; vÞ ¼ V X j ~ T ðvÞeij S ðvÞV S xr S þ X j ~ T ðvÞeij AðvÞV A xr A ð10Þ All the wave modes propagate independently in the structure. The final waveform will be the superposition of all the propagating waves and will have the contribution from each Lamb mode. Insertion of damage effects into the analytical model Figure 6 shows the pitch-catch active sensing method for damage detection: the T-PWAS transducer generates ultrasonic guided waves which propagate into the structure, interact with structural damage at x ¼ xd , carry the damage information with them, and are picked up by the R-PWAS transducer at x ¼ xr . To model the damage effect on Lamb wave propagation, we consider the damage as a new wave source at x ¼ xd , and we add mode conversion and nonlinear sources at the damage location through damage interaction coefficients. The predictive analytical model for Lamb wave interaction with damage is constructed in frequency domain in the following steps: Step 1. This step is identical to step 1 of the pristine case. Perform Fourier transform of the time-domain excitation signal VT ðtÞ to obtain the frequency-domain ~T ðvÞ. excitation spectrum, V Step 2. Calculate the frequency-domain structural transfer function up to the damage location, Gðxd ; vÞ. The structure transfer function Gðxd ; vÞ is similar to equation (3) of previous section, only that x ¼ xd , that is Gðxd ; vÞ ¼ S ðvÞeij S xd þ AðvÞeij A xd ð11Þ Step 3. Multiply the structural transfer function by frequency-domain excitation signal to obtain the frequency-domain signal at the damage location, that ~D ðxd ; vÞ ¼ Gðxd ; vÞ V ~T ðvÞ. Hence, the signal at is, V the damage location is ~D ðxd ; vÞ ¼ S ðvÞV ~T ðvÞeijS xd þ AðvÞV ~T ðvÞeijA xd ð12Þ V This signal could be decomposed into symmetric and antisymmetric components ~ S ðxd ; vÞ ¼ S ðvÞV ~T ðvÞeijS xd V D ð13Þ ~ A ðxd ; vÞ ¼ AðvÞV ~T ðvÞeijA xd V D ð14Þ Step 4. The wave signal at the damage location takes the damage information by considering transmission, reflection, mode conversion, and higher harmonics. Each of these addition phenomena is modeled as a new wave source at the damage location using damage interaction coefficients (Figure 7). We distinguish two damage interaction types: (a) linear and (b) nonlinear, as discussed next. Linear damage interaction. Wave transmission, reflection, and mode conversion are realized by using complexamplitude damage interaction coefficients. Our notations are as follows: we use three letters to describe the interaction phenomena, with the first letter denoting the incident wave type, the second letter standing for resulting wave type, and the third letter meaning propagation direction (transmission/reflection). For instance, symmetric-symmetric-transmission (SST) means the incident symmetric waves transmitted as symmetric waves, while symmetric-antisymmetric-transmission (SAT) means incident symmetric waves transmitted and mode converted to antisymmetric waves. Thus, the complex-amplitude damage interaction coefficient CSST eiuSST denotes the transmitted symmetric mode generated by incident symmetric mode with magnitude Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Shen and Giurgiutiu 7 Figure 7. Modeling wave transmission, reflection, mode conversion, and higher harmonics components. CSST and phase uSST . Similarly, CSAT eiuSAT represents the transmitted antisymmetric mode generated by incident symmetric mode with magnitude CSAT and phase uSAT . These coefficients are determined by the features of the damage and are to be imported into the WFR model. where M is the number of higher harmonics considered. For linear interaction with damage, M equals to one. Step 6. Perform inverse Fourier transform to obtain the time-domain receiver sensing signal Nonlinear damage interaction. The center frequency of waves arriving at the damage location can be obtained from equations (13) and (14) as vc . The second and third higher harmonics act as wave sources with center frequencies of 2vc and 3vc , respectively. Modeling of higher harmonics is achieved by moving the frequencydomain signal at the damage location to the right-hand side of the frequency axis by vc and 2vc , that is, ~D ðxd ; v vc Þ ~3D ðxd ; vÞ ¼ ~2D ðxd ; vÞ ¼ V and V V ~D ðxd ; v 2vc Þ represent the second and third higher V harmonics nonlinear wave source. The nonlinear damage interaction coefficients are defined in the same way as the linear ones. For instance, the complex-amplitude damage interaction coefficient M M eiuSST denotes the Mth higher harmonics transCSST mitted symmetric mode generated by incident symM and phase uM . metric mode with magnitude CSST SST It should be noted that the above analysis only considers S0 and A0 modes. But the principle could be easily extended to higher modes (S1, A1, etc.). The difficulty with extending to higher modes will be on defining the increasing number of transmission, reflection, and mode conversion coefficients. For each excited Lamb mode, the interaction with damage may result in more mode conversion possibilities. In this study, the WFR has been designed to simulate (a) multi-mode (S0, A0, S1, A1) Lamb waves propagation in pristine plates and (b) fundamental modes (S0 and A0) Lamb wave interaction with damage. Step 5. The guided waves from the new wave sources created at the damage location propagate through the rest of the structure and arrive at the R-PWAS. The received wave signal is calculated in frequency domain as The analytical representation of this process was coded in MATLAB and resulted in the GUI called WFR shown in Figure 8. ~ R ðxd ; xr ; vÞ ¼ V þ m h X M¼1 m h X VR ðxd ; xr ; tÞ ¼ IFFT fV~R ðxd ; xr ; vÞg WFR interface and main functions i M iuM ~ S ðxd ; vÞ þ C M eiuMAST V ~ A ðxd ; vÞ eijS ðxr xd Þ CSST e SST V MD AST MD M M CAAT eiuAAT ð16Þ i ~ A ðxd ; vÞ þ C M eiuMSAT V ~ S ðxd ; vÞ eijA ðxr xd Þ V MD SAT MD M¼1 Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 ð15Þ 8 Structural Health Monitoring Figure 8. Main GUI of WaveFormRevealer. GUI: graphic user interface. WFR allows users to control several parameters: structure material properties, PWAS size, location of sensors, location of damage, damage type (linear/nonlinear damage of various severities), and excitation signal (frequencies, count numbers, signal mode excitation, arbitrary waveform type, etc.). Dual display of waveforms allows for the sensing signals to be shown at two different sensor locations. For instance, Figure 8 shows two receiver waveforms at locations x1 ¼ 0 mm and x2 ¼ 500 mm as measured from the transmitter (in this case x1 ¼ 0 mm means that R-PWAS-1 collocated with the T-PWAS). Thus, PWAS-1 shows the reflections from damage, and PWAS-2 shows the signal modified after passing through the damage. Users are able to conduct fast parametric studies with WFR. It may take several hours for commercial finite element software to obtain an acceptable-accuracy solution for high-frequency, long distance propagating waves, but it takes only several seconds to obtain the same solution with WFR. Besides analytical waveform solutions, the WFR can also provide users with wave speed dispersion curves, tuning curves, frequency components of received wave packets, structure transfer function, and so on. All the calculated results are fully available to the user, and could be saved in Excel files by clicking the ‘‘SAVE’’ button. Figure 9 shows a case study for Lamb wave propagation of a 100 kHz tone burst in a 1-mm-thick aluminum plate. Figure 9(a) shows the dispersion curves; Figure 9(b) shows the excitation spectrum overlap with the S0 and A0 tuning curves. Figure 9(c) shows the spectra of the S0 and A0 packets displaying frequency shifts to the right and to the left, respectively, due to the interaction between excitation spectrum and the tuning curves. Figure 9(d) shows the structure transfer function Gðx; vÞ. Besides the main interface, WFR has two subinterfaces shown in Figure 10: (a) damage information platform and (b) guided wave spatial propagation solver. The damage information platform allows users to input the damage location and damage interaction coefficients. For example, SST represents the magnitude of transmitted S0 mode generated by an incoming Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Shen and Giurgiutiu (a) 9 4 (b) x 10 2 c (m/s) A3 S3 Normalized Amplitude A2 A1 1.5 S1 S2 1 S0 0.5 A0 0 6000 4000 f (kHz) 2000 1.2 8000 1 0.8 S0 0.6 0.4 A0 0.2 0 10000 0 200 600 400 Frequency kHz 800 1000 100 200 300 Frequency kHz 400 500 (d) 1.5 Excitation 100 kHz 1 0.8 A0 0.6 Magnitude Normalized Amplitude (c) 0 1.2 Excitation S0 0.4 1 0.5 0.2 0 0 50 100 150 Frequency kHz 200 0 0 Figure 9. Calculation of various quantities in Lamb wave propagation: (a) wave speed dispersion curve, (b) tuning curve, (c) frequency contents of received wave packets, and (d) structure transfer function. S0 mode, whereas SAT and phi-SAT represent the magnitude and phase of the transmitted A0 mode resulting from the mode conversion of an incoming S0 mode. The values of these damage interaction coefficients are not calculated by the WFR. This gives the users the freedom to define their own specific problem. For instance, a particular type of damage (plastic zone, fatigue, cracks) with certain degree of severity will have different interaction characteristics with the interrogating guided waves. These coefficients may be determined experimentally or calculated through other methods (analytical, FEM, BEM, etc.). Among all the above methods, FEM approach shows good results for obtaining the interaction coefficients of arbitrary shaped damage. Successful examples and details can be found in Velichko and Wilcox20,21 and Moreau et al.22 In an example presented later in this study, we used a trial-and-error approach to tune the WFR coefficients to the data obtained from experiments and finite element simulations. The spatial propagation solver is like a B-scan. Using the analytical procedure, we obtain the time-domain waveform solution at various locations along the structure. Thus, the time-domain waveform solutions of a sequence of points along the wave propagation path are obtained. If we select the sequence of solution points fine enough, a time-spatial domain solution of the wave field is obtained. The spatial solution of wave field at a particular instance in time is available as shown in Figure 10(b). After the time-spatial solution of wave field is obtained, we can do the frequency–wavenumber analysis23 to see the wave components of the signal (Figure 11). These will be illustrated in the case studies discussed later in this article. Case studies Linear interaction with damage of selective Lamb wave modes WFR allows users to select single mode (S0 and A0) or multi-mode (S0 and A0) to be excited into the structure. Three test cases were conducted: (a) incident S0 wave linear interaction with damage, (b) incident A0 wave linear interaction with damage, and (c) combined S0 and A0 waves linear interaction with damage. The test case setup is shown in Figure 12. The T-PWAS and R-PWAS are placed 600 mm away from each other on a 1-mm-thick aluminum 2024-T3 plate. The damage is Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 10 Structural Health Monitoring Figure 10. User interfaces: (a) damage information platform and (b) guided wave spatial propagation solver. placed 200 mm from the T-PWAS. A 5-count Hanning window modulated tone burst centered at 100 kHz is used as the excitation. The time-domain and the time– frequency domain signals of the test cases are shown in Figure 13. Figure 13 shows that new wave packets appear due to the interaction between interrogation Lamb waves and damage. Incident S0 wave will generate A0 wave from mode conversion at the damage, whereas incident A0 wave will generate S0 wave from mode conversion at the damage. However, from the time-frequency analysis, it could be observed that after linear interaction, the frequency spectrum of the waves still center around the excitation frequency 100 kHz. Nonlinear interaction with damage of selective Lamb wave modes As test cases for nonlinear interaction between Lamb waves and damage, three simulations were carried out: Figure 11. Frequency–wavenumber display window. Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Shen and Giurgiutiu 11 T-PWAS R-PWAS Damage 200 mm 400 mm Figure 12. Test case setup for pitch-catch Lamb wave interaction with damage. T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor. Time domain signal from WFR 0.5 0 -0.5 Transmitted S0 -1 50 100 150 400 300 200 100 200 250 300 350 400 450 500 0 50 100 150 Time domain signal from WFR 1 A0 Excitation 0.5 Mode converted S0 from incident A0 wave -0.5 Transmitted A0 -1 100 (c) 150 200 250 300 350 400 450 0 Transmitted S0 50 100 New packet from mode conversion at damage 150 200 250 300 500 400 450 500 400 450 500 200 100 0 50 100 350 150 200 250 300 350 Transmitted A0 500 400 300 200 100 0 0 450 Time (microsecond) Time-frequency domain signal 0.5 -1 400 300 500 S0 and A0 Excitation -0.5 350 400 Time (microsecond) Time domain signal from WFR 1 300 500 0 50 250 Time-frequency domain signal 0 0 200 Time (microsecond) Time (microsecond) Frequency (kHz) Normalized amplitude 500 0 0 (b) Normalized amplitude Time-frequency domain signal Mode converted A0 from incident S0 wave S0 Excitation Frequency (kHz) 1 Frequency (kHz) Normalized amplitude (a) 400 450 500 Time (microsecond) 0 50 100 150 200 250 300 350 Time (microsecond) Figure 13. Simulation of linear interaction between Lamb waves and damage: (a) S0 mode excitation, (b) A0 mode excitation, and (c) S0 and A0 modes excitation. It should be noted that no higher harmonics are observed. WFR: WaveFormRevealer. (a) incident S0 wave nonlinear interaction with damage, (b) incident A0 wave nonlinear interaction with damage, and (c) combined S0 and A0 waves nonlinear interaction with damage. The test case setup is the same as shown in Figure 12, only the interaction with damage is nonlinear. The time signals and the time-frequency analysis of the test cases are shown in Figure 14. It can be observed in Figure 14 that after nonlinear interaction with the damage, the waveforms become distorted and contain distinctive nonlinear higher harmonics. For S0 waves which are less dispersive at the given frequency range, the nonlinear higher harmonics stay inside the wave packet. However, for A0 waves which are dispersive at the given frequency range, the higher harmonic components travel faster, leading the way and may escape from the fundamental wave packet. Experimental verifications Multi-mode Lamb wave propagation in a pristine plate In our study, two PWAS transducers were mounted on a 3.17-mm-thick aluminum 7075-T6 plate. Figure 15 shows the experiment setup. The T-PWAS sends out ultrasonic guided waves into the structure. The guided waves, that is, Lamb waves propagate in the plate, Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 12 Structural Health Monitoring Time domain signal from WFR S0 Excitation 0 -0.5 Transmitted S0 with distortion -1 0 50 100 150 200 500 300 350 400 450 Dispersive A0 mode generated from mode conversion from S0 wave 400 300 200 100 0 250 500 0 50 100 1 A0 Excitation 0 Mode converted S0 with distortion from incident A0 -1 0 50 100 150 200 250 Transmitted A0 350 1 400 450 300 0 50 100 150 200 250 300 Transmitted A0 350 450 500 nonlinear higher harmonics 100 0 50 100 150 200 250 300 350 400 450 500 400 450 500 Time-frequency domain signal 0 -1 400 Time (microsecond) 0.5 Transmitted S0 New packet from mode with distortion conversion at damage 350 200 Distinctive 500 S0 and A0 Excitation -0.5 300 Dispersive transmitted A0 wave 400 Time (microsecond) Time domain signal from WFR (c) 250 500 0 300 200 Time-frequency domain signal 0.5 -0.5 150 Time (microsecond) Time (microsecond) Time domain signal from WFR Frequency (kHz) Normalized amplitude Mode converted A0 from incident S0 wave 0.5 (b) Normalized amplitude Time-frequency domain signal Frequency (kHz) 1 400 450 500 Frequency (kHz) Normalized amplitude (a) 500 400 300 200 100 0 0 50 100 150 200 250 300 350 Time (microsecond) Time (microsecond) Figure 14. Simulation of nonlinear interaction between Lamb waves and damage: (a) S0 mode excitation, (b) A0 mode excitation, and (c) S0 and A0 modes excitation. It should be noted that distinctive higher harmonics are observed. WFR: WaveFormRevealer. T-PWAS 303 mm propagation path R-PWAS Figure 15. Experiment setup for multi-mode Lamb wave propagation. WFR: WaveFormRevealer. undergoing dispersion and are picked up by the R-PWAS. The Lamb waves are multi-modal, hence several wave packets appear in the received signal. Agilent 33120A Arbitrary Waveform Generator is used to generate 3-count Hanning window modulated tone burst excitations. A Tektronix Digital Oscilloscope is used to record the experimental waveforms. The excitation frequency is increased from 300 to 600 kHz. Corresponding plate material, thickness, PWAS size, and sensing location information is input into the WFR. The analytical waveforms of various frequencies are obtained. Figure 16 shows the comparison between analytical solution from WFR and experimental data. It can be observed that at 300 kHz, only S0 and A0 modes exist. The WFR solution matches well with experimental data. At 450 kHz, S0 mode becomes more dispersive; besides S0 and A0 modes, A1 mode starts to pick up with highly dispersive feature. At 600 kHz, S0, A0, and A1 modes exist simultaneously. The simulation results and the experimental data have slight differences due to the fact that 1D analytical formulas and pin force excitation assumptions are used in this study. To further validate WFR predictions, we also conducted 2D FEM simulation with pin force excitation (1D Lamb wave propagation simulation). Figure 17 shows the comparison between WFR and FEM simulations. It can be observed that the 300 and 450 kHz waveforms match very well between WFR and FEM. Signals of 600 kHz also have reasonably good agreement. It should be noted, even for 1D Lamb wave propagation simulation, that the 600 kHz wave computation requires considerably small element size and time marching step. The FEM simulation for such high-frequency, short-wavelength situation is becoming prohibitive due to the heavy consumption of computation time and computer resources. On the contrary, WFR only requires several seconds to obtain the same results due to its highly efficient analytical formulation. The guided wave spatial propagation solver in WFR is used to obtain the time–space wave field (B-scan) as shown in Figure 18(a). The frequency–wavenumber analysis is conducted next, as shown in Figure 18(b). Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Shen and Giurgiutiu 13 1 0.5 A0 S0 300 kHz Experiment WFR 0 -0.5 Normalized Amplitude -1 0 20 40 60 80 100 120 140 1 160 180 Experiment WFR 0.5 450 kHz 0 -0.5 S0 dispersive wave 0 20 40 A1 dispersive wave A0 -1 60 80 100 120 140 1 160 180 Experiment WFR 0.5 600 kHz 0 -0.5 S0 dispersive wave A0 -1 0 20 40 60 80 100 A1 dispersive wave 120 140 160 180 Time (microsecond) Figure 16. Comparison between WFR and experiment for multi-mode Lamb wave propagation in a pristine 3.17-mm aluminum plate. WFR: WaveFormRevealer. The 600 kHz case is used as an example. From the Bscan, S0, A0, and A1 wave components can be observed. Frequency–wavenumber analysis gives very clear information on the wave mode components of the wave field. Transmitted S0 wave (S0-T), A0 wave (A0-T), and A1 wave (A1-T) are clearly noticed in Figure 18(b). Linear interaction between Lamb waves and damage Pitch-catch mode. Figure 19 shows the experimental specimen (3.17-mm-thick Aluminum-7075-T6 plate), with PWAS #3 used as the transmitter (T-PWAS) and PWAS #4 used as the receiver (R-PWAS). A notch (h1 ¼ 2:5 mm; d1 ¼ 0:25 mm) is machined on the plate, 143.5 mm from the T-PWAS. The wave propagation path from T-PWAS to R-PWAS is 303 mm. The 3count Hanning window modulated tone burst signals with center frequencies varying from 150 to 300 kHz are used as the excitation. S0 and A0 waves are transmitted by the T-PWAS. At the notch, S0 waves will be transmitted as S0 waves and also will be mode converted to transmitted A0 waves. A0 waves will be transmitted as A0 waves and also will be mode converted to transmitted S0 waves. All these transmitted waves will propagate along the rest of the structure and be picked up by the R-PWAS. The damage interaction coefficients are physically determined by the size, severity, type of the damage. In this study, we used a trial-and-error approach to tune the WFR damage interaction coefficients to the data obtained from the experiments. The adjusted damage interaction coefficients which gave best match with experiments for 150 kHz excitation case are shown in Table 1. Figure 20 shows the WFR simulation results compared with experiments. It can be noticed that the analytical waveforms agree well with experimental data. A new wave packet is generated due to mode conversion at the notch. Pulse-echo mode. Figure 21 shows the experimental setup for pulse-echo active sensing method. The same specimen is used, with an R-PWAS bounded side by Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 14 Structural Health Monitoring 1 0.5 WFR FEM 300 kHz 0 -0.5 Normalized Amplitude -1 0 20 40 60 80 100 120 140 160 1 0.5 180 WFR FEM 450 kHz 0 -0.5 -1 0 20 40 60 80 100 120 140 160 1 0.5 180 WFR FEM 600 kHz 0 -0.5 -1 0 20 40 60 80 100 120 140 160 180 Time (microsecond) Figure 17. Comparison between WFR and FEM for multi-mode Lamb wave propagation in a pristine 3.17-mm aluminum plate. WFR: WaveFormRevealer; FEM: finite element method. (a) (b) Time-space wave fieldSignal (B-scan) Time-Space Domain Frequency-wavenumber analysis Frequency-Wavenumber Domain Signal 2000 300 A0-T 250 1000 200 150 Wavenumber Space (mm) A1 S0 A0 100 S0-T A1-T 0 A1-R S0-R -1000 50 0 S1-T S1-R A0-R -2000 0 20 40 60 80 100 Time (microsecond) 120 0 500 1000 1500 Frequency (kHz) 2000 Figure 18. (a) Time–space wave field (B-scan) and (b) frequency–wavenumber analysis from WFR. WFR: WaveFormRevealer. side to the T-PWAS. The 3-count Hanning window modulated tone burst signals with the center frequency of 95.5 kHz is used as the excitation. Guided Lamb waves generated by the T-PWAS will propagate into the structure, reach the notch, and be reflected back as echoes. At the notch, S0 waves will be reflected as S0 Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Shen and Giurgiutiu 15 Figure 19. Experiment for Lamb wave linear interaction with a notch (pitch-catch mode). T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor. 1 Experiment WFR S0 0.5 A0 150 kHz 0 -0.5 New packet from mode conversion (S0+A0) -1 0 1 Normalized Amplitude 0.5 20 40 60 80 100 120 140 160 180 Experiment WFR 200 kHz 0 -0.5 New packet from mode conversion (S0+A0) -1 0 1 0.5 20 40 60 80 100 120 140 160 180 60 80 100 120 140 160 180 60 80 100 120 140 160 180 Experiment WFR 250 kHz 0 -0.5 -1 0 1 20 40 Experiment WFR 0.5 300 kHz 0 -0.5 -1 0 20 40 Time (microsecond) Figure 20. Comparison between WFR simulations and experiments for Lamb wave interaction with a notch in pitch-catch mode. WFR: WaveFormRevealer. Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 16 Structural Health Monitoring T-PWAS 143.5 mm Notch R-PWAS Figure 21. Experiment for Lamb wave linear interaction with a notch (pulse-echo mode). T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor. waves and also will be mode converted to reflected A0 waves. A0 waves will be reflected as A0 waves and also will be mode converted to reflected S0 waves. All the echoes will reach the R-PWAS and be picked up. The adjusted damage interaction coefficients which gave best match with the experiment are shown in Table 2. Figure 22 shows the WFR simulation result compared with the experiment. The reflected S0 and A0 wave packets could be observed. The new waves between S0 and A0 wave packets are from mode Table 1. Damage interaction coefficients for pitch-catch mode. 1 CSST 0.55 u1SST 230 Magnitude coefficient Value (normalized) Phase coefficient Value (°) 1 CSAT 0.11 u1SAT 30 1 CAAT 0.8 u1AAT 0 1 CAST 0.06 u1AST 30 Table 2. Damage interaction coefficients for pulse-echo mode. 1 CSSR 0.2 u1SSR 60 Normalized amplitude Magnitude coefficient Value (normalized) Phase coefficient Value (°) 1 CSAR 0.04 u1SAR 60 1 CAAR 0.12 u1AAR 260 Direct waves 0.2 1 CASR 0.04 u1ASR 60 ReflectedS0 conversion at the notch. The analytical simulation matches the experiment data. Differences are noticed: first, the direct waves have a phase shift due to the fact that the R-PWAS and T-PWAS are some distance away from each other, while in our analytical model, we consider them to be at the same location; second, the boundary reflections are present and mixed with the weak echoes from the notch in the experiment, but in our model, the boundary reflections are not considered. Figure 23 shows the results from WFR spatial propagation solver. The wave transmission, reflection, and mode conversion can be clearly noticed in both the Bscan and frequency–wavenumber analysis. It is apparent that the wave field contains transmitted S0 and A0 modes, and reflected S0 and A0 modes. Nonlinear interaction between Lamb waves and damage A guided wave pitch-catch method may be used to interrogate a plate with a breathing crack which opens and closes under tension and compression.6,24 The ultrasonic waves generated by the T-PWAS propagate into the structure, interact with the breathing crack, acquire nonlinear features, and are picked up by the RPWAS. This process is shown in Figure 24. The nonlinear interaction between Lamb waves and the breathing crack will introduce nonlinear higher harmonics into the interrogation waves. A multi-physics transient finite element model was used to simulate the Lamb wave interaction with a nonlinear breathing crack. The damage interaction coefficients obtained from fitting the FEM solution (Table 3) were input into the WFR simulator. Figure 25 shows the comparison between FEM and the WFR analytical solution. It is noticed that the FEM results and the analytical solution agree very well because the damage interaction coefficients were fitted to the FEM solution. The time-domain waveforms show nonlinear characteristics of noticeable nonlinear ReflectedA0 Experiment WFR 0.1 0 -0.1 -0.2 Wave packet from mode conversion 0 50 100 150 Boundary reflections 200 250 300 Time (microsecond) Figure 22. Comparison between WFR simulations and experiments for Lamb wave interaction with a notch in pulse-echo mode. WFR: WaveFormRevealer. Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 Shen and Giurgiutiu (a) 17 (b) 300 AST AAT 250 1500 Forward transmission through damage 1000 A0-T Wavenumber Space (mm) SST 200 SAT 150 Damage 100 500 S0-T 0 -500 S0-R A0-R SAR 50 ASR SSR 0 0 -1000 AAR -1500 50 100 Time (microsecond) Backward reflection from damage 0 100 200 300 400 Frequency (kHz) 500 Figure 23. (a) Time–space domain solution (B-scan) and (b) frequency–wavenumber analysis from WFR with transmission, reflection, and mode conversion damage effects. WFR: WaveFormRevealer. Generation of higher harmonics …….. Breathing crack T-PWAS R-PWAS Figure 24. Pitch-catch method for the detection of breathing crack; the mode conversion at the crack is illustrated by the two arrows. T-PWAS: transmitter piezoelectric wafer active sensor; R-PWAS: receiver piezoelectric wafer active sensor. Table 3. Nonlinear interaction coefficients. Magnitude coefficient Value (normalized) Phase coefficient Value (°) 1 CSST 0.900 u1SST 0 1 CSAT 0.420 u1SAT 100 1 CAAT 0.820 u1AAT 235 1 CAST 0.100 u1AST 90 2 CSST 0.082 u2SST 0 distortion in S0 packet and zigzags in the new packet. The frequency spectrums show distinctive nonlinear higher harmonics (200 and 300 kHz). Since we only consider up to the third higher harmonic in this case study, the frequency domain of analytical solution shows only the first three peaks, while the finite element solution have even higher harmonics. But the solution up to the third higher harmonics is accurate enough to render an acceptable waveform in time domain. The guided wave spatial propagation solver in WFR was used to obtain the time–space wave field. Figure 26 shows the time–space wave field and frequency– wavenumber analysis of Lamb wave interaction with nonlinear breathing crack. 2 CSAT 0.100 u2SAT 0 2 CAAT 0.050 u2AAT 120 2 CAST 0.110 u2AST 90 3 CSST 0.032 u3SST 0 3 CSAT 0.038 u3SAT 0 3 CAAT 0.005 u3AAT 0 3 CAST 0.025 u3AST 0 Transmission, reflection, and mode conversion phenomena at the damage can be clearly noticed. The frequency–wavenumber analysis reveals the wave components during the interaction process. The wave field contains transmitted S0 and A0 waves and reflected S0 and A0 waves. Nonlinear higher harmonics can be observed at 200 kHz. The WFR-guided wave spatial propagation solver can provide the spatial wave pattern at any instance of time. The spatial waveforms at 0, 25, 50, 75, 100, 125, 150, 175, and 200 ms are displayed in Figure 27. The spatial waveforms shows (a) Lamb waves propagating into the structure at T ¼ 25 ms, (b) Lamb modes separating into distinct packets at T ¼ 50 ms, (c) Lamb wave Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 18 Structural Health Monitoring (a) 1 Normalized amplitude Waveform with zigzags A0 0.5 WFR 0 -0.5 New wave packet from nonlinear interaction Distorted S0 FEM -1 0 50 100 150 200 250 300 350 Time (microsecond) (b) (d) FEM 0 FEM FEM 0 10 0 10 Magnitude 10 (c) -5 10 10 -5 WFR WFR 0 200 400 600 800 1000 0 200 400 600 WFR 1000 0 800 200 600 800 1000 Frequency (kHz) Frequency (kHz) Frequency (kHz) 400 Figure 25. (a) Comparison between finite element simulation (FEM) and analytical simulation (WFR), (b) frequency spectrum of S0 packet, (c) frequency spectrum of new packet and (d) frequency spectrum of A0 packet. WFR: WaveFormRevealer; FEM: finite element method. (a) (b) 500 Nonlinear higher harmonics Forward transmission 1000 through damage 300 Damage 200 Wave Number Space (mm) 400 A0-T 500 S0-T 0 S0-R -500 A0-R Backward reflection -1000 from damage 100 0 1500 0 50 100 150 200 Time (microsecond) 250 300 -1500 0 50 100 150 200 250 Frequency (kHz) Figure 26. (a) Time–space wave field (B-scan) and (b) frequency–wavenumber analysis from WFR. WFR: WaveFormRevealer. Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 300 350 Shen and Giurgiutiu 19 1 T = 0 μs Damage ↓ 0 -1 0 50 100 150 1 200 250 300 350 400 450 500 T = 25 μs Damage ↓ 0 Waves propagating into the structure; S0 and A0 mixed together -1 0 1 50 100 A0 150 S0 0 0 50 100 150 1 250 300 400 450 500 T = 50 μs ↓ S0 wave interacting with damage 200 250 300 350 400 ↓ 450 500 T = 75 μs Transmitted S0 Damage 0 350 Damage S0 and A0 model separation -1 Normalized Amplitude 200 Transmitted A0 from mode conversion -1 0 1 50 100 150 Reflected A0 from mode conversion 0 Reflected S0 -1 0 50 250 Damage 100 150 0 50 100 150 200 250 100 300 350 400 450 500 T = 125 μs New wave packet from mode conversion containing both S0 and A0 modes 200 250 300 350 400 450 500 T = 150 μs ↓ Reflected A0 50 500 Transmitted S0 Damage 0 0 450 T = 100 μs ↓ Reflected S0 from mode conversion -1 400 ↓ Reflected S0 from mode conversion 1 350 Damage 0 -1 300 A0 wave interacting with damage New wave packet from mode conversion Reflected A0 from mode conversion 1 200 150 1 Transmitted A0 200 250 300 350 400 450 500 T = 200 μs Damage ↓ 0 Transmitted A0 Reflected A0 -1 0 50 100 150 200 250 300 Space (mm) 350 400 Figure 27. Spatial wave propagation of Lamb wave interaction with breathing crack (calculated using WFR). WFR: WaveFormRevealer. Downloaded from shm.sagepub.com at UNIVERSITY OF SOUTH CAROLINA on May 14, 2014 450 500 20 Structural Health Monitoring packets interaction with the damage also at T ¼ 50 ms, and (d) wave transmission, reflection, mode conversion, and nonlinear distortion of waveforms at various instances (T ¼ 75; 100; 125; 150; and 200 ms). Summary, conclusions, and future work Summary should be built to simulate wave attenuation in waveguides. Boundary reflection and damage effects in 2D wave propagation should be investigated. Attempts for simulating guided wave propagation in composite structures should be made using WFR. Declaration of conflicting interests The authors declare that there is no conflict of interest. In this study, we presented the WFR—an analytical framework and predictive tool for the simulation of guided Lamb wave interaction with damage. The theory of inserting damage effects into the analytical model was addressed, including wave transmission, reflection, mode conversion, and nonlinear higher harmonics components. The analytical model was coded into MATLAB, and the WFR GUI was developed to obtain fast predictive waveforms for arbitrary combinations of sensors, structural properties, and damage. Main functions of WFR were introduced, including the calculation for dispersion curves, tuning curves, frequency spectrum of sensing signal, plate transfer function, time–space domain waveforms with damage effects, frequency–wavenumber analysis, and the capability of considering arbitrary user defined excitation signals. Test cases were carried out. Experimental verifications were presented. The predictive solution from WFR agreed well with experiments and finite element simulations. WFR can be downloaded from: http:// www.me.sc.edu/Research/lamss/html/software.html. Conclusion The WFR was capable of calculating dispersion curves, tuning curves, frequency components of wave packets, and structural transfer function. It could be used to obtain time–space domain waveforms with damage effects and frequency–wavenumber analysis. WFR could provide fast predictive solutions for multi-mode Lamb wave propagation and interaction with linear/ nonlinear damage. The solutions compared well with experiments and finite element simulations. It was also found that computational time savings of several orders of magnitude are obtained by using the analytical model WFR instead of FEM methods. WFR allowed users to conduct fast parametric studies with their own designed materials, geometries, and excitations. Future work Rational methods for determining damage interaction coefficient values need to be found (not trial and error). Work should be carried out to extend the analysis to 2D wave propagation (three-dimensional (3D) FEM and 2D WFR). The 2D WFR with damping effect Funding The following funding supports for this study are thankfully acknowledged: Office of Naval Research # N00014-11-10271, Dr. Ignacio Perez, Technical Representative; Air Force Office of Scientific Research # FA9550-11-1-0133, Dr. David Stargel, Program Manager. References 1. Graff KF. Wave motion in elastic solids. New York: Dover publications, Inc., 1991. 2. Rose JL. Ultrasonic waves in solid media. Cambridge: Cambridge University Press, 1999. 3. Giurgiutiu V. Structural health monitoring with piezoelectric wafer active sensors. Oxford, UK: Elsevier Academic Press, 2007. 4. Jhang KY. Nonlinear ultrasonic techniques for nondestructive assessment of micro damage in material: a review. Int J Precis Eng Man 2009; 10: 123–135. 5. Dutta D, Sohn H and Harries KA. A nonlinear acoustic technique for crack detection in metallic structures. Struct Health Monit 2009; 8: 251–262. 6. Shen Y and Giurgiutiu V. 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